1 Introduction

In forming our queuing model for SGH PAC, the role and aim in this section is to extract distributions for these two variables:

  1. Queueing time. This is the variable that indicates the amount of time taken between the arrival of a patient and the consult with the anaesthesiologist.
  2. Service time. This is the variable that indicates the amount of time taken in a consult with the anaesthesiologist.

The data given to us is from 2019 to 2021 for both planned appointments and walk-in appointments. However, we will only focus on data from 2020 onwards due to SGH’s claim that the 2019 data may not be as reliable.

2 Assumptions

Acknowledging that the real world data may not be ideal, we would like to lay down some assumptions regarding the data that will help with the model-fitting.

  1. There is no exact data for queueing time. Therefore, it will be assumed that the queueing time is the total duration that the patient is in PAC subtracted by the actual consult duration. Implicitly, the other PAC stations (e.g. Radiology) where patients are assessed are considered as a “queue” in this case. For our purposes, this decision is justified as the project focuses on the anaesthesiologists and the consults thereof.
  2. That any service time below 15 minutes is considered a data error. We must also acknowledge the assumption that any data above that is fully accurate to a reasonable degree, except when in the case that it is unrealistically high.
  3. Seasonality near the Chinese New Year, which is usually in February. This information is provided to us by SGH through their past experience.

3 Methodology

This is a mostly quantitative analysis. The data will be cleaned, meaning that any data error will be removed, and appropriate data types shall be applied to each variable. Data from 2020 and 2021 will then be aggregated into two tibbles (in essence, a better version of a data frame for R), one for planned appointments and walk-in appointments. From the master tibble, they will be split based on both seasonality (February versus the rest of the months) and ASA Score (measure of health in a pre-surgery assessment). There will be sixteen tibbles formed through this split, knowing that ASA Scores range from 1 to 4. However, there may not be enough data to form a distribution for the ASA Score of 4 due to its rarity.

The sixteen tibbles will be analysed using descriptive statistics in the beginning, and then we will try to fit likely distributions to it. Ideally, the distributions will all be exponential. The main test that will be done to corroborate the distribution and its parameters will be the chi-squared test at \(\alpha\) = 0.05.

4 Data Cleaning

4.1 Raw Data

4.1.1 2020 Planned Appointments

## # A tibble: 6 x 21
##   `PAC Date` `Visit Time` Identifier  Age   `Visit dept. OU` `Procedure Code`
##   <chr>      <chr>        <chr>       <chr> <chr>            <chr>           
## 1 1/7/20     8:00         44013879ZGG 63    H&N - ENT        SJ802T          
## 2 1/7/20     8:00         44013201IGJ 71    Orthop Surg      SB741S          
## 3 1/7/20     8:00         44013138HIG 36    Orthop Surg      SB700H          
## 4 1/7/20     8:00         44013339ZGG 47    Gynaecology      SI710V          
## 5 1/7/20     8:30         44013492ZAZ 3     Otolary          SM701E          
## 6 1/7/20     8:30         44013516ZGH 69    Orthop Surg      SB810K          
## # … with 15 more variables: No Show / Attended <chr>, Listing Date <chr>,
## #   No. of days between PAC date and listing date <chr>, Surgery Date <chr>,
## #   Lead time to Surgery <chr>, Admit Specialty <chr>, ASA Score <chr>,
## #   PAC Registration Time <chr>, PAC End Time <chr>,
## #   PAC Consult Duration <chr>, TOSP Table Code <chr>,
## #   Complexity of Surgery <chr>, TPS Patient (Yes/No) <chr>,
## #   TPS Patient (Recruited before PAC / Onsite) <chr>, X21 <lgl>

4.1.2 2020 Walk-In Appointments

## # A tibble: 6 x 20
##   `PAC Date` `Visit Time` Identifier  Age   `Visit dept. OU` `Procedure Code`
##   <chr>      <chr>        <chr>       <chr> <chr>            <chr>           
## 1 1/7/20     8:00         44013585ENB 22    Orthop Surg      SB703K          
## 2 1/7/20     10:00        44013980AKI 35    Orthop Surg      SB840H          
## 3 1/7/20     11:00        44013923ABC 66    VASCULAR SUR     SD817A          
## 4 1/7/20     9:30         44013272ANF 67    Colorec Surg     SF802C          
## 5 1/7/20     10:00        44013680FGG 85    Colorec Surg     SF802C          
## 6 1/7/20     14:30        44013335III 47    Gynaecology      SI725U          
## # … with 14 more variables: No Show / Attended <chr>, Listing Date <chr>,
## #   No. of days between PAC date and listing date <chr>, Surgery Date <chr>,
## #   Lead time to Surgery <chr>, Admit Specialty <chr>, ASA Score <chr>,
## #   PAC Registration Time <chr>, PAC End Time <chr>,
## #   PAC Consult Duration <chr>, TOSP Table Code <chr>,
## #   Complexity of Surgery <chr>, TPS Patient (Yes/No) <chr>,
## #   TPS Patient (Recruited before PAC / Onsite) <chr>

4.1.3 2021 Planned Appointments

## # A tibble: 6 x 20
##   `PAC Date` `Visit Time` Identifier  Age   `Visit dept. OU` `Procedure Code`
##   <chr>      <time>       <chr>       <chr> <chr>            <chr>           
## 1 4/1/21     08:00        44200586JIG 23    Urology          SJ802T          
## 2 4/1/21     08:00        44200015FDG 57    GI/BARIATRIC     SB715K          
## 3 4/1/21     08:00        44200377JGH 88    Orthop Surg      SB816S          
## 4 4/1/21     08:00        44200448ZNC 75    GI/BARIATRIC     SB727S          
## 5 4/1/21     08:00        44200247HKC #N/A  EYE              #N/A            
## 6 4/1/21     08:00        44200829FEF 74    HRM              SB810K          
## # … with 14 more variables: No Show / Attended <chr>, Listing Date <chr>,
## #   No. of days between PAC date and listing date <chr>, Surgery Date <chr>,
## #   Lead time to Surgery <chr>, Admit Specialty <chr>, ASA Score <chr>,
## #   PAC Registration Time <chr>, PAC End Time <chr>,
## #   PAC Consult Duration <chr>, TOSP Table Code <chr>,
## #   Complexity of Surgery <chr>, TPS Patient (Yes/No) <chr>,
## #   TPS Patient (Recruited before PAC / Onsite) <chr>

4.1.4 2021 Walk-In Appointments

## # A tibble: 6 x 21
##   `PAC Date` `Visit Time` Identifier  Age   `Visit dept. OU` `Procedure Code`
##   <chr>      <chr>        <chr>       <chr> <chr>            <chr>           
## 1 4/1/21     13:30        44200185FGB 62    Colorec Surg     SF836A          
## 2 4/1/21     15:00        44200222HGZ 37    Gynaecology      SI843U          
## 3 4/1/21     8:00         44200230EID 27    Gynaecology      SF836A          
## 4 5/1/21     14:30        44201892EEG 41    GI/BARIATRIC     SF703S          
## 5 5/1/21     11:30        44201820DOH 85    VASCULAR SUR     SD712A          
## 6 5/1/21     13:30        44201147ANI 76    VASCULAR SUR     SD821A          
## # … with 15 more variables: No Show / Attended <chr>, Listing Date <chr>,
## #   No. of days between PAC date and listing date <chr>, Surgery Date <chr>,
## #   Lead time to Surgery <chr>, Admit Specialty <chr>, ASA Score <chr>,
## #   PAC Registration Time <chr>, PAC End Time <chr>,
## #   PAC Consult Duration <chr>, TOSP Table Code <chr>,
## #   Complexity of Surgery <chr>, TPS Patient (Yes/No) <chr>,
## #   TPS Patient (Recruited before PAC / Onsite) <chr>, X21 <lgl>

4.2 Cleaned Data

4.2.1 Planned Appointments Aggregate

## # A tibble: 6 x 3
##   asa   queue_time consult_duration
##   <chr>      <int>            <int>
## 1 1             52               33
## 2 2             69               46
## 3 1             72               28
## 4 1             28               20
## 5 2             90               30
## 6 2            413               17
Descriptive statistics for planned appointments
ASA Score Queue Time Consult Duration
Min. :1.000 Min. : 13.0 Min. : 15.00
1st Qu.:2.000 1st Qu.: 94.0 1st Qu.: 21.00
Median :2.000 Median :131.0 Median : 29.00
Mean :1.822 Mean :144.6 Mean : 35.18
3rd Qu.:2.000 3rd Qu.:180.0 3rd Qu.: 42.00
Max. :4.000 Max. :556.0 Max. :319.00

Histogram for queuing time for appointments in PAC

Histogram for consult duration for appointments in PAC

Seeing the histogram for consult duration, it might be a bit more useful to think of it as the amount of time beyond 15 minutes the consult will take. Hence, we will translate the consult duration by 15 minutes to the left.

Histogram for adjusted consult duration for appointments in PAC

4.2.2 Walk-In Appointments Aggregate

## # A tibble: 6 x 3
##   asa   queue_time consult_duration
##   <chr>      <dbl>            <int>
## 1 1             80               17
## 2 3            189               41
## 3 3            136               74
## 4 2            334               23
## 5 2            291               68
## 6 1             59               18
Descriptive statistics for walk-in appointments
ASA Score Queue Time Consult Duration
Min. :1.000 Min. : 16.0 Min. : 15.00
1st Qu.:1.000 1st Qu.: 95.0 1st Qu.: 20.00
Median :2.000 Median :139.0 Median : 28.00
Mean :1.794 Mean :160.9 Mean : 34.61
3rd Qu.:2.000 3rd Qu.:206.0 3rd Qu.: 40.00
Max. :3.000 Max. :585.0 Max. :185.00

Similarly to the planned appointments, the three figures below describe the queue time and consult duration in histograms.

Histogram for queuing time for walk-ins in PAC

Histogram for consult duration for walk-ins in PAC

Histogram for adjusted consult duration for walk-ins in PAC

4.2.3 Appointments Aggregate

## # A tibble: 6 x 3
##   asa   queue_time consult_duration
##   <chr>      <dbl>            <int>
## 1 1             52               33
## 2 2             69               46
## 3 1             72               28
## 4 1             28               20
## 5 2             90               30
## 6 2            413               17

Histogram for queuing time in PAC

Histogram for consult duration in PAC

Histogram for adjusted consult duration in PAC

5 Finding Distributions

5.1 Planned and Walk-In Appointments

5.1.1 Queue Time

5.1.1.1 ASA 1

## summary statistics
## ------
## min:  13   max:  585 
## median:  115 
## mean:  129.4893 
## estimated sd:  67.28252 
## estimated skewness:  1.78132 
## estimated kurtosis:  8.385513
5.1.1.1.1 Testing for Lognormal Distribution

## [[1]]
## Goodness-of-fit statistics
##                              1-mme-lnorm
## Kolmogorov-Smirnov statistic   0.0179067
## Cramer-von Mises statistic     0.0603894
## Anderson-Darling statistic     0.6085893
## 
## Goodness-of-fit criteria
##                                1-mme-lnorm
## Akaike's Information Criterion    23851.54
## Bayesian Information Criterion    23862.92
## 
## [[2]]
## Fitting of the distribution ' lnorm ' by matching moments 
## Parameters : 
##          estimate
## meanlog 4.7441447
## sdlog   0.4887809
## Loglikelihood:  -11923.77   AIC:  23851.54   BIC:  23862.92
Chi-Squared Test for Lognormal Distribution with MeanLog 4.7441447 and SdLog 0.4887809
Pass Error Critical Value
Not rejected 3.790103 9.487729
5.1.1.1.2 Testing for Gamma Distribution

## [[1]]
## Goodness-of-fit statistics
##                              1-mme-gamma
## Kolmogorov-Smirnov statistic  0.05277488
## Cramer-von Mises statistic    1.55534146
## Anderson-Darling statistic   12.68223134
## 
## Goodness-of-fit criteria
##                                1-mme-gamma
## Akaike's Information Criterion    23981.55
## Bayesian Information Criterion    23992.94
## 
## [[2]]
## Fitting of the distribution ' gamma ' by matching moments 
## Parameters : 
##         estimate
## shape 3.70562271
## rate  0.02861722
## Loglikelihood:  -11988.78   AIC:  23981.55   BIC:  23992.94
Chi-Squared Test for Gamma Distribution with Shape 3.70562271 and Rate 0.02861722
Pass Error Critical Value
Rejected 39.74268 9.487729

5.1.1.2 ASA 2

## summary statistics
## ------
## min:  27   max:  569 
## median:  137 
## mean:  150.9156 
## estimated sd:  71.03786 
## estimated skewness:  1.191075 
## estimated kurtosis:  4.97947
5.1.1.2.1 Testing for Gamma Distribution

## [[1]]
## Goodness-of-fit statistics
##                              1-mme-gamma
## Kolmogorov-Smirnov statistic  0.02898534
## Cramer-von Mises statistic    1.13535433
## Anderson-Darling statistic   10.55945776
## 
## Goodness-of-fit criteria
##                                1-mme-gamma
## Akaike's Information Criterion    69662.87
## Bayesian Information Criterion    69676.35
## 
## [[2]]
## Fitting of the distribution ' gamma ' by matching moments 
## Parameters : 
##         estimate
## shape 4.51396193
## rate  0.02991051
## Loglikelihood:  -34829.43   AIC:  69662.87   BIC:  69676.35
Chi-Squared Test for Gamma Distribution with Shape 4.51396193 and Rate 0.02991051
Pass Error Critical Value
Not rejected 17.96091 18.30704
5.1.1.2.2 Testing for Lognormal Distribution

## [[1]]
## Goodness-of-fit statistics
##                              1-mme-lnorm
## Kolmogorov-Smirnov statistic  0.02405473
## Cramer-von Mises statistic    0.69879880
## Anderson-Darling statistic    4.89547405
## 
## Goodness-of-fit criteria
##                                1-mme-lnorm
## Akaike's Information Criterion    69527.77
## Bayesian Information Criterion    69541.25
## 
## [[2]]
## Fitting of the distribution ' lnorm ' by matching moments 
## Parameters : 
##          estimate
## meanlog 4.9166666
## sdlog   0.4473345
## Loglikelihood:  -34761.88   AIC:  69527.77   BIC:  69541.25
Chi-Squared Test for Lognormal Distribution with MeanLog 4.9116591 and SdLog 0.4419957
Pass Error Critical Value
Rejected 29.33012 19.67514

5.1.1.3 ASA 3

## summary statistics
## ------
## min:  47   max:  495 
## median:  137 
## mean:  154.6518 
## estimated sd:  73.04713 
## estimated skewness:  1.272831 
## estimated kurtosis:  5.050589
5.1.1.3.1 Testing for Gamma Distribution

## [[1]]
## Goodness-of-fit statistics
##                              1-mme-gamma
## Kolmogorov-Smirnov statistic  0.05181065
## Cramer-von Mises statistic    0.35534652
## Anderson-Darling statistic    2.48916670
## 
## Goodness-of-fit criteria
##                                1-mme-gamma
## Akaike's Information Criterion    6256.461
## Bayesian Information Criterion    6265.117
## 
## [[2]]
## Fitting of the distribution ' gamma ' by matching moments 
## Parameters : 
##        estimate
## shape 4.4903452
## rate  0.0290352
## Loglikelihood:  -3126.23   AIC:  6256.461   BIC:  6265.117
Chi-Squared Test for Gamma Distribution with Shape 4.4903452 and Rate 0.0290352
Pass Error Critical Value
Not rejected 13.37659 18.30704

5.1.2 Consult Duration

Adjusted version, meaning that the consult duration was shifted by 15 minutes, unless otherwise stated.

5.1.2.1 ASA 1

## summary statistics
## ------
## min:  0   max:  204 
## median:  11 
## mean:  16.0456 
## estimated sd:  18.30492 
## estimated skewness:  3.079084 
## estimated kurtosis:  18.48056
5.1.2.1.1 Testing for Lognormal Distribution

The lognormal distribution requires numbers above 0. Hence, the shifting in this case is by 14, such that the minimum is 1.

## [[1]]
## Goodness-of-fit statistics
##                              1-mme-lnorm
## Kolmogorov-Smirnov statistic  0.09808718
## Cramer-von Mises statistic    2.65276066
## Anderson-Darling statistic   38.40126539
## 
## Goodness-of-fit criteria
##                                1-mme-lnorm
## Akaike's Information Criterion    16941.03
## Bayesian Information Criterion    16952.41
## 
## [[2]]
## Fitting of the distribution ' lnorm ' by matching moments 
## Parameters : 
##          estimate
## meanlog 2.4525327
## sdlog   0.8756247
## Loglikelihood:  -8468.513   AIC:  16941.03   BIC:  16952.41
Chi-Squared Test for Lognormal Distribution with MeanLog 2.4525327 and SdLog 0.8756247
Pass Error Critical Value
Not rejected 9.605517 15.50731
5.1.2.1.2 Testing for Gamma Distribution

## [[1]]
## Goodness-of-fit statistics
##                              1-mme-gamma
## Kolmogorov-Smirnov statistic  0.07372338
## Cramer-von Mises statistic    2.59119338
## Anderson-Darling statistic           Inf
## 
## Goodness-of-fit criteria
##                                1-mme-gamma
## Akaike's Information Criterion        -Inf
## Bayesian Information Criterion        -Inf
## 
## [[2]]
## Fitting of the distribution ' gamma ' by matching moments 
## Parameters : 
##         estimate
## shape 0.76873108
## rate  0.04790915
## Loglikelihood:  Inf   AIC:  -Inf   BIC:  -Inf
Chi-Squared Test for Gamma Distribution with Shape 0.76873108 and Rate 0.04790915
Pass Error Critical Value
Rejected 55.54124 14.06714
5.1.2.1.3 Testing for Exponential Distribution

## [[1]]
## Goodness-of-fit statistics
##                               1-mme-exp
## Kolmogorov-Smirnov statistic 0.05380757
## Cramer-von Mises statistic   0.70027975
## Anderson-Darling statistic          Inf
## 
## Goodness-of-fit criteria
##                                1-mme-exp
## Akaike's Information Criterion  16561.06
## Bayesian Information Criterion  16566.75
## 
## [[2]]
## Fitting of the distribution ' exp ' by matching moments 
## Parameters : 
##        estimate
## rate 0.06232238
## Loglikelihood:  -8279.528   AIC:  16561.06   BIC:  16566.75
Chi-Squared Test for Exponential Distribution with Rate 0.06232238
Pass Error Critical Value
Rejected 42.66968 15.50731

5.1.2.2 ASA 2

## summary statistics
## ------
## min:  1   max:  305 
## median:  16 
## mean:  21.73269 
## estimated sd:  21.55089 
## estimated skewness:  2.857424 
## estimated kurtosis:  18.73142
5.1.2.2.1 Testing for Lognormal Distribution

The lognormal distribution requires numbers above 0. Hence, the shifting in this case is by 14, such that the minimum is 1.

## [[1]]
## Goodness-of-fit statistics
##                               1-mme-lnorm
## Kolmogorov-Smirnov statistic   0.09597839
## Cramer-von Mises statistic     9.78075237
## Anderson-Darling statistic   146.12106517
## 
## Goodness-of-fit criteria
##                                1-mme-lnorm
## Akaike's Information Criterion    51935.91
## Bayesian Information Criterion    51949.39
## 
## [[2]]
## Fitting of the distribution ' lnorm ' by matching moments 
## Parameters : 
##          estimate
## meanlog 2.7364665
## sdlog   0.8274675
## Loglikelihood:  -25965.95   AIC:  51935.91   BIC:  51949.39
Chi-Squared Test for Lognormal Distribution with MeanLog 2.7364665 and SdLog 0.8274675
Pass Error Critical Value
Rejected 45.55516 5.991465
5.1.2.2.2 Testing for Gamma Distribution

## [[1]]
## Goodness-of-fit statistics
##                              1-mme-gamma
## Kolmogorov-Smirnov statistic  0.05101785
## Cramer-von Mises statistic    2.85874812
## Anderson-Darling statistic           Inf
## 
## Goodness-of-fit criteria
##                                1-mme-gamma
## Akaike's Information Criterion        -Inf
## Bayesian Information Criterion        -Inf
## 
## [[2]]
## Fitting of the distribution ' gamma ' by matching moments 
## Parameters : 
##         estimate
## shape 0.92565787
## rate  0.04464725
## Loglikelihood:  Inf   AIC:  -Inf   BIC:  -Inf
Chi-Squared Test for Gamma Distribution with Shape 0.92565787 and Rate 0.04464725
Pass Error Critical Value
Rejected 15.25287 3.841459
5.1.2.2.3 Testing for Exponential Distribution

## [[1]]
## Goodness-of-fit statistics
##                               1-mme-exp
## Kolmogorov-Smirnov statistic 0.03533173
## Cramer-von Mises statistic   1.08195066
## Anderson-Darling statistic          Inf
## 
## Goodness-of-fit criteria
##                                1-mme-exp
## Akaike's Information Criterion  50438.72
## Bayesian Information Criterion  50445.46
## 
## [[2]]
## Fitting of the distribution ' exp ' by matching moments 
## Parameters : 
##      estimate
## rate 0.048233
## Loglikelihood:  -25218.36   AIC:  50438.72   BIC:  50445.46
Chi-Squared Test for Exponential Distribution with Rate 0.048233
Pass Error Critical Value
Not rejected 5.7713 5.991465

5.1.2.3 ASA 3

## summary statistics
## ------
## min:  1   max:  292 
## median:  22 
## mean:  30.45536 
## estimated sd:  30.72181 
## estimated skewness:  2.914877 
## estimated kurtosis:  17.01842
5.1.2.3.1 Testing for Lognormal Distribution

The lognormal distribution requires numbers above 0. Hence, the shifting in this case is by 14, such that the minimum is 1.

## [[1]]
## Goodness-of-fit statistics
##                              1-mme-lnorm
## Kolmogorov-Smirnov statistic  0.06304681
## Cramer-von Mises statistic    0.35976253
## Anderson-Darling statistic    6.46302707
## 
## Goodness-of-fit criteria
##                                1-mme-lnorm
## Akaike's Information Criterion    4986.764
## Bayesian Information Criterion    4995.420
## 
## [[2]]
## Fitting of the distribution ' lnorm ' by matching moments 
## Parameters : 
##          estimate
## meanlog 3.0657644
## sdlog   0.8372545
## Loglikelihood:  -2491.382   AIC:  4986.764   BIC:  4995.42
Chi-Squared Test for Lognormal Distribution with MeanLog 3.0657644 and SdLog 0.8372545
Pass Error Critical Value
Not rejected 5.50356 12.59159
5.1.2.3.2 Testing for Gamma Distribution

## [[1]]
## Goodness-of-fit statistics
##                              1-mme-gamma
## Kolmogorov-Smirnov statistic  0.08827393
## Cramer-von Mises statistic    0.73718487
## Anderson-Darling statistic           Inf
## 
## Goodness-of-fit criteria
##                                1-mme-gamma
## Akaike's Information Criterion        -Inf
## Bayesian Information Criterion        -Inf
## 
## [[2]]
## Fitting of the distribution ' gamma ' by matching moments 
## Parameters : 
##         estimate
## shape 0.92089709
## rate  0.03126416
## Loglikelihood:  Inf   AIC:  -Inf   BIC:  -Inf
Chi-Squared Test for Gamma Distribution with Shape 0.92089709 and Rate 0.03126416
Pass Error Critical Value
Not rejected 5.842161 5.991465
5.1.2.3.3 Testing for Exponential Distribution

## [[1]]
## Goodness-of-fit statistics
##                               1-mme-exp
## Kolmogorov-Smirnov statistic 0.06998185
## Cramer-von Mises statistic   0.40160340
## Anderson-Darling statistic          Inf
## 
## Goodness-of-fit criteria
##                                1-mme-exp
## Akaike's Information Criterion  4910.821
## Bayesian Information Criterion  4915.149
## 
## [[2]]
## Fitting of the distribution ' exp ' by matching moments 
## Parameters : 
##        estimate
## rate 0.03394968
## Loglikelihood:  -2454.41   AIC:  4910.821   BIC:  4915.149
Chi-Squared Test for Exponential Distribution with Rate 0.03394968
Pass Error Critical Value
Not rejected 4.483978 7.814728

5.2 Planned Appointments

5.2.1 Queue Time

5.2.1.1 ASA 1

## summary statistics
## ------
## min:  13   max:  556 
## median:  114 
## mean:  127.7265 
## estimated sd:  64.52942 
## estimated skewness:  1.79333 
## estimated kurtosis:  8.60614
5.2.1.1.1 Testing for Lognormal Distribution

## [[1]]
## Goodness-of-fit statistics
##                              1-mme-lnorm
## Kolmogorov-Smirnov statistic  0.01654383
## Cramer-von Mises statistic    0.06195682
## Anderson-Darling statistic    0.63112144
## 
## Goodness-of-fit criteria
##                                1-mme-lnorm
## Akaike's Information Criterion    21557.21
## Bayesian Information Criterion    21568.41
## 
## [[2]]
## Fitting of the distribution ' lnorm ' by matching moments 
## Parameters : 
##          estimate
## meanlog 4.7362772
## sdlog   0.4766837
## Loglikelihood:  -10776.6   AIC:  21557.21   BIC:  21568.41
Chi-Squared Test for Lognormal Distribution with MeanLog 4.7362772 and SdLog 0.4766837
Pass Error Critical Value
Not rejected 5.192523 9.487729
5.2.1.1.2 Testing for Gamma Distribution

## [[1]]
## Goodness-of-fit statistics
##                              1-mme-gamma
## Kolmogorov-Smirnov statistic  0.05008021
## Cramer-von Mises statistic    1.31740467
## Anderson-Darling statistic   10.95928126
## 
## Goodness-of-fit criteria
##                                1-mme-gamma
## Akaike's Information Criterion    21678.92
## Bayesian Information Criterion    21690.12
## 
## [[2]]
## Fitting of the distribution ' gamma ' by matching moments 
## Parameters : 
##         estimate
## shape 3.91979880
## rate  0.03068901
## Loglikelihood:  -10837.46   AIC:  21678.92   BIC:  21690.12
Chi-Squared Test for Gamma Distribution with Shape 3.9197988 and Rate 0.03068901
Pass Error Critical Value
Rejected 22.16605 7.814728

5.2.1.2 ASA 2

## summary statistics
## ------
## min:  27   max:  535 
## median:  137 
## mean:  149.8057 
## estimated sd:  69.58894 
## estimated skewness:  1.183166 
## estimated kurtosis:  4.995085
5.2.1.2.1 Testing for Gamma Distribution

## [[1]]
## Goodness-of-fit statistics
##                              1-mme-gamma
## Kolmogorov-Smirnov statistic  0.02925145
## Cramer-von Mises statistic    0.92462179
## Anderson-Darling statistic    9.11450436
## 
## Goodness-of-fit criteria
##                                1-mme-gamma
## Akaike's Information Criterion    64053.78
## Bayesian Information Criterion    64067.10
## 
## [[2]]
## Fitting of the distribution ' gamma ' by matching moments 
## Parameters : 
##        estimate
## shape 4.6350196
## rate  0.0309402
## Loglikelihood:  -32024.89   AIC:  64053.78   BIC:  64067.1
Chi-Squared Test for Gamma Distribution with Shape 4.6350196 and Rate 0.0309402
Pass Error Critical Value
Not rejected 20.84311 21.02607
5.2.1.2.2 Testing for Lognormal Distribution

## [[1]]
## Goodness-of-fit statistics
##                              1-mme-lnorm
## Kolmogorov-Smirnov statistic  0.02588754
## Cramer-von Mises statistic    0.74568786
## Anderson-Darling statistic    5.00402712
## 
## Goodness-of-fit criteria
##                                1-mme-lnorm
## Akaike's Information Criterion    63930.79
## Bayesian Information Criterion    63944.11
## 
## [[2]]
## Fitting of the distribution ' lnorm ' by matching moments 
## Parameters : 
##          estimate
## meanlog 4.9116591
## sdlog   0.4419957
## Loglikelihood:  -31963.4   AIC:  63930.79   BIC:  63944.11
Chi-Squared Test for Lognormal Distribution with MeanLog 4.9116591 and SdLog 0.4419957
Pass Error Critical Value
Rejected 38.91663 22.36203

5.2.1.3 ASA 3

## summary statistics
## ------
## min:  47   max:  480 
## median:  136 
## mean:  151.883 
## estimated sd:  69.55987 
## estimated skewness:  1.204725 
## estimated kurtosis:  4.808147
5.2.1.3.1 Testing for Gamma Distribution

## [[1]]
## Goodness-of-fit statistics
##                              1-mme-gamma
## Kolmogorov-Smirnov statistic   0.0467720
## Cramer-von Mises statistic     0.2687312
## Anderson-Darling statistic     1.8789186
## 
## Goodness-of-fit criteria
##                                1-mme-gamma
## Akaike's Information Criterion    5691.367
## Bayesian Information Criterion    5699.847
## 
## [[2]]
## Fitting of the distribution ' gamma ' by matching moments 
## Parameters : 
##         estimate
## shape 4.77692467
## rate  0.03145134
## Loglikelihood:  -2843.683   AIC:  5691.367   BIC:  5699.847
Chi-Squared Test for Gamma Distribution with Shape 4.77692467 and Rate 0.03145134
Pass Error Critical Value
Not rejected 13.61228 14.06714

5.2.2 Consult Duration

Adjusted version, meaning that the consult duration was shifted by 15 minutes, unless otherwise stated.

5.2.2.1 ASA 1

## summary statistics
## ------
## min:  0   max:  204 
## median:  11 
## mean:  16.10571 
## estimated sd:  18.33906 
## estimated skewness:  3.129485 
## estimated kurtosis:  19.16101
5.2.2.1.1 Testing for Lognormal Distribution

The lognormal distribution requires numbers above 0. Hence, the shifting in this case is by 14, such that the minimum is 1.

## [[1]]
## Goodness-of-fit statistics
##                              1-mme-lnorm
## Kolmogorov-Smirnov statistic   0.0954679
## Cramer-von Mises statistic     2.2509975
## Anderson-Darling statistic    34.5996657
## 
## Goodness-of-fit criteria
##                                1-mme-lnorm
## Akaike's Information Criterion    15441.94
## Bayesian Information Criterion    15453.14
## 
## [[2]]
## Fitting of the distribution ' lnorm ' by matching moments 
## Parameters : 
##         estimate
## meanlog 2.456952
## sdlog   0.874598
## Loglikelihood:  -7718.972   AIC:  15441.94   BIC:  15453.14
Chi-Squared Test for Lognormal Distribution with MeanLog 2.456952 and SdLog 0.874598
Pass Error Critical Value
Not rejected 16.66601 16.91898
5.2.2.1.2
5.2.2.1.3 Testing for Gamma Distribution

## [[1]]
## Goodness-of-fit statistics
##                              1-mme-gamma
## Kolmogorov-Smirnov statistic  0.07696044
## Cramer-von Mises statistic    2.50299959
## Anderson-Darling statistic           Inf
## 
## Goodness-of-fit criteria
##                                1-mme-gamma
## Akaike's Information Criterion        -Inf
## Bayesian Information Criterion        -Inf
## 
## [[2]]
## Fitting of the distribution ' gamma ' by matching moments 
## Parameters : 
##        estimate
## shape 0.7716552
## rate  0.0479119
## Loglikelihood:  Inf   AIC:  -Inf   BIC:  -Inf
Chi-Squared Test for Gamma Distribution with Shape 0.7716552 and Rate 0.0479119
Pass Error Critical Value
Not rejected 13.26114 14.06714
5.2.2.1.4 Testing for Exponential Distribution

## [[1]]
## Goodness-of-fit statistics
##                               1-mme-exp
## Kolmogorov-Smirnov statistic 0.05511022
## Cramer-von Mises statistic   0.63235091
## Anderson-Darling statistic          Inf
## 
## Goodness-of-fit criteria
##                                1-mme-exp
## Akaike's Information Criterion  15088.46
## Bayesian Information Criterion  15094.06
## 
## [[2]]
## Fitting of the distribution ' exp ' by matching moments 
## Parameters : 
##        estimate
## rate 0.06208978
## Loglikelihood:  -7543.231   AIC:  15088.46   BIC:  15094.06
Chi-Squared Test for Exponential Distribution with Rate 0.06208978
Pass Error Critical Value
Rejected 34.53983 16.91898

5.2.2.2 ASA 2

## summary statistics
## ------
## min:  1   max:  305 
## median:  16 
## mean:  21.73328 
## estimated sd:  21.38637 
## estimated skewness:  2.89165 
## estimated kurtosis:  19.4802
5.2.2.2.1 Testing for Lognormal Distribution

The lognormal distribution requires numbers above 0. Hence, the shifting in this case is by 14, such that the minimum is 1.

## [[1]]
## Goodness-of-fit statistics
##                               1-mme-lnorm
## Kolmogorov-Smirnov statistic   0.09452814
## Cramer-von Mises statistic     8.75709084
## Anderson-Darling statistic   133.97976693
## 
## Goodness-of-fit criteria
##                                1-mme-lnorm
## Akaike's Information Criterion    47926.62
## Bayesian Information Criterion    47939.94
## 
## [[2]]
## Fitting of the distribution ' lnorm ' by matching moments 
## Parameters : 
##          estimate
## meanlog 2.7402942
## sdlog   0.8228612
## Loglikelihood:  -23961.31   AIC:  47926.62   BIC:  47939.94
Chi-Squared Test for Lognormal Distribution with MeanLog 2.7402942 and SdLog 0.8228612
Pass Error Critical Value
Rejected 21.42409 7.814728
5.2.2.2.2 Testing for Gamma Distribution

## [[1]]
## Goodness-of-fit statistics
##                              1-mme-gamma
## Kolmogorov-Smirnov statistic   0.0522794
## Cramer-von Mises statistic     2.6976054
## Anderson-Darling statistic           Inf
## 
## Goodness-of-fit criteria
##                                1-mme-gamma
## Akaike's Information Criterion        -Inf
## Bayesian Information Criterion        -Inf
## 
## [[2]]
## Fitting of the distribution ' gamma ' by matching moments 
## Parameters : 
##        estimate
## shape 0.9400198
## rate  0.0453387
## Loglikelihood:  Inf   AIC:  -Inf   BIC:  -Inf
Chi-Squared Test for Gamma Distribution with Shape 0.9400198 and Rate 0.0453387
Pass Error Critical Value
Not rejected 5.40839 5.991465
5.2.2.2.3 Testing for Exponential Distribution

## [[1]]
## Goodness-of-fit statistics
##                               1-mme-exp
## Kolmogorov-Smirnov statistic 0.03887442
## Cramer-von Mises statistic   1.26676599
## Anderson-Darling statistic          Inf
## 
## Goodness-of-fit criteria
##                                1-mme-exp
## Akaike's Information Criterion  46528.28
## Bayesian Information Criterion  46534.94
## 
## [[2]]
## Fitting of the distribution ' exp ' by matching moments 
## Parameters : 
##        estimate
## rate 0.04823165
## Loglikelihood:  -23263.14   AIC:  46528.28   BIC:  46534.94
Chi-Squared Test for Exponential Distribution with Rate 0.04823165
Pass Error Critical Value
Not rejected 11.10927 12.59159

5.2.2.3 ASA 3

## summary statistics
## ------
## min:  1   max:  292 
## median:  22 
## mean:  30.81287 
## estimated sd:  31.43551 
## estimated skewness:  2.907392 
## estimated kurtosis:  16.71943
5.2.2.3.1 Testing for Lognormal Distribution

The lognormal distribution requires numbers above 0. Hence, the shifting in this case is by 14, such that the minimum is 1.

## [[1]]
## Goodness-of-fit statistics
##                              1-mme-lnorm
## Kolmogorov-Smirnov statistic   0.0636774
## Cramer-von Mises statistic     0.3279006
## Anderson-Darling statistic     5.7790015
## 
## Goodness-of-fit criteria
##                                1-mme-lnorm
## Akaike's Information Criterion    4577.690
## Bayesian Information Criterion    4586.171
## 
## [[2]]
## Fitting of the distribution ' lnorm ' by matching moments 
## Parameters : 
##          estimate
## meanlog 3.0717530
## sdlog   0.8440134
## Loglikelihood:  -2286.845   AIC:  4577.69   BIC:  4586.171
Chi-Squared Test for Lognormal Distribution with MeanLog 3.071753 and SdLog 0.8440134
Pass Error Critical Value
Not rejected 4.778335 7.814728
5.2.2.3.2 Testing for Gamma Distribution

## [[1]]
## Goodness-of-fit statistics
##                              1-mme-gamma
## Kolmogorov-Smirnov statistic  0.09086619
## Cramer-von Mises statistic    0.72208232
## Anderson-Darling statistic           Inf
## 
## Goodness-of-fit criteria
##                                1-mme-gamma
## Akaike's Information Criterion        -Inf
## Bayesian Information Criterion        -Inf
## 
## [[2]]
## Fitting of the distribution ' gamma ' by matching moments 
## Parameters : 
##         estimate
## shape 0.90118456
## rate  0.03022804
## Loglikelihood:  Inf   AIC:  -Inf   BIC:  -Inf
Chi-Squared Test for Gamma Distribution with Shape 0.90118456 and Rate 0.03022804
Pass Error Critical Value
Not rejected 8.379399 9.487729
5.2.2.3.3 Testing for Exponential Distribution

## [[1]]
## Goodness-of-fit statistics
##                               1-mme-exp
## Kolmogorov-Smirnov statistic 0.06771132
## Cramer-von Mises statistic   0.34491026
## Anderson-Darling statistic          Inf
## 
## Goodness-of-fit criteria
##                                1-mme-exp
## Akaike's Information Criterion  4511.208
## Bayesian Information Criterion  4515.449
## 
## [[2]]
## Fitting of the distribution ' exp ' by matching moments 
## Parameters : 
##        estimate
## rate 0.03354257
## Loglikelihood:  -2254.604   AIC:  4511.208   BIC:  4515.449
Chi-Squared Test for Exponential Distribution with Rate 0.03354257
Pass Error Critical Value
Not rejected 6.057588 11.0705

5.3 Walk-In Appointments

5.3.1 Queue Time

5.3.1.1 ASA 1

## summary statistics
## ------
## min:  16   max:  585 
## median:  127 
## mean:  147.3503 
## estimated sd:  88.83391 
## estimated skewness:  1.380647 
## estimated kurtosis:  5.790429
5.3.1.1.1 Testing for Lognormal Distribution

## [[1]]
## Goodness-of-fit statistics
##                              1-mme-lnorm
## Kolmogorov-Smirnov statistic   0.0596964
## Cramer-von Mises statistic     0.1086712
## Anderson-Darling statistic     0.7974009
## 
## Goodness-of-fit criteria
##                                1-mme-lnorm
## Akaike's Information Criterion    2269.386
## Bayesian Information Criterion    2275.953
## 
## [[2]]
## Fitting of the distribution ' lnorm ' by matching moments 
## Parameters : 
##          estimate
## meanlog 4.8384769
## sdlog   0.5555817
## Loglikelihood:  -1132.693   AIC:  2269.386   BIC:  2275.953
Chi-Squared Test for Lognormal Distribution with MeanLog 4.8384769 and SdLog 0.5555817
Pass Error Critical Value
Not rejected 5.394277 9.487729
5.3.1.1.2 Testing for Gamma Distribution

## [[1]]
## Goodness-of-fit statistics
##                              1-mme-gamma
## Kolmogorov-Smirnov statistic  0.04825587
## Cramer-von Mises statistic    0.08126044
## Anderson-Darling statistic    0.60684980
## 
## Goodness-of-fit criteria
##                                1-mme-gamma
## Akaike's Information Criterion    2265.978
## Bayesian Information Criterion    2272.544
## 
## [[2]]
## Fitting of the distribution ' gamma ' by matching moments 
## Parameters : 
##         estimate
## shape 2.76537723
## rate  0.01876737
## Loglikelihood:  -1130.989   AIC:  2265.978   BIC:  2272.544
Chi-Squared Test for Gamma Distribution with Shape 2.76537723 and Rate 0.01876737
Pass Error Critical Value
Not rejected 2.535864 7.814728

5.3.1.2 ASA 2

## summary statistics
## ------
## min:  28   max:  569 
## median:  143 
## mean:  164.1196 
## estimated sd:  85.41705 
## estimated skewness:  1.072297 
## estimated kurtosis:  4.166928
5.3.1.2.1 Testing for Gamma Distribution

## [[1]]
## Goodness-of-fit statistics
##                              1-mme-gamma
## Kolmogorov-Smirnov statistic  0.04899709
## Cramer-von Mises statistic    0.20490028
## Anderson-Darling statistic    1.22771882
## 
## Goodness-of-fit criteria
##                                1-mme-gamma
## Akaike's Information Criterion    5576.171
## Bayesian Information Criterion    5584.539
## 
## [[2]]
## Fitting of the distribution ' gamma ' by matching moments 
## Parameters : 
##         estimate
## shape 3.69937258
## rate  0.02254071
## Loglikelihood:  -2786.086   AIC:  5576.171   BIC:  5584.539
Chi-Squared Test for Gamma Distribution with Shape 3.69937258 and Rate 0.02254071
Pass Error Critical Value
Not rejected 21.08321 22.36203
5.3.1.2.2 Testing for Lognormal Distribution

## [[1]]
## Goodness-of-fit statistics
##                              1-mme-lnorm
## Kolmogorov-Smirnov statistic  0.03889669
## Cramer-von Mises statistic    0.18078797
## Anderson-Darling statistic    1.48333130
## 
## Goodness-of-fit criteria
##                                1-mme-lnorm
## Akaike's Information Criterion    5577.235
## Bayesian Information Criterion    5585.603
## 
## [[2]]
## Fitting of the distribution ' lnorm ' by matching moments 
## Parameters : 
##         estimate
## meanlog 4.980962
## sdlog   0.489148
## Loglikelihood:  -2786.618   AIC:  5577.235   BIC:  5585.603
Chi-Squared Test for Lognormal Distribution with MeanLog 4.980962 and SdLog 0.489148
Pass Error Critical Value
Not rejected 18.26444 22.36203

5.3.1.3 ASA 3

## summary statistics
## ------
## min:  49   max:  495 
## median:  173 
## mean:  184.8723 
## estimated sd:  99.8424 
## estimated skewness:  1.078161 
## estimated kurtosis:  3.911473
5.3.1.3.1 Testing for Gamma Distribution

## [[1]]
## Goodness-of-fit statistics
##                              1-mme-gamma
## Kolmogorov-Smirnov statistic  0.11069041
## Cramer-von Mises statistic    0.07231369
## Anderson-Darling statistic    0.42905363
## 
## Goodness-of-fit criteria
##                                1-mme-gamma
## Akaike's Information Criterion    556.3347
## Bayesian Information Criterion    560.0350
## 
## [[2]]
## Fitting of the distribution ' gamma ' by matching moments 
## Parameters : 
##         estimate
## shape 3.50311077
## rate  0.01894881
## Loglikelihood:  -276.1673   AIC:  556.3347   BIC:  560.035
Chi-Squared Test for Gamma Distribution with Shape 3.50311077 and Rate 0.01894881
Pass Error Critical Value
Not rejected 7.495405 14.06714

5.3.2 Consult Duration

Adjusted version, meaning that the consult duration was shifted by 15 minutes, unless otherwise stated.

5.3.2.1 ASA 1

## summary statistics
## ------
## min:  0   max:  103 
## median:  10 
## mean:  15.43655 
## estimated sd:  17.98986 
## estimated skewness:  2.553924 
## estimated kurtosis:  11.12355
5.3.2.1.1 Testing for Lognormal Distribution

The lognormal distribution requires numbers above 0. Hence, the shifting in this case is by 14, such that the minimum is 1.

## [[1]]
## Goodness-of-fit statistics
##                              1-mme-lnorm
## Kolmogorov-Smirnov statistic   0.1281475
## Cramer-von Mises statistic     0.4791926
## Anderson-Darling statistic     4.4066376
## 
## Goodness-of-fit criteria
##                                1-mme-lnorm
## Akaike's Information Criterion    1502.302
## Bayesian Information Criterion    1508.868
## 
## [[2]]
## Fitting of the distribution ' lnorm ' by matching moments 
## Parameters : 
##          estimate
## meanlog 2.4071329
## sdlog   0.8858606
## Loglikelihood:  -749.1508   AIC:  1502.302   BIC:  1508.868
Chi-Squared Test for Lognormal Distribution with MeanLog 2.4071329 and SdLog 0.8858606
Pass Error Critical Value
Not rejected 0.8173222 7.814728
5.3.2.1.2 Testing for Gamma Distribution

## [[1]]
## Goodness-of-fit statistics
##                              1-mme-gamma
## Kolmogorov-Smirnov statistic  0.09339454
## Cramer-von Mises statistic    0.16358727
## Anderson-Darling statistic           Inf
## 
## Goodness-of-fit criteria
##                                1-mme-gamma
## Akaike's Information Criterion        -Inf
## Bayesian Information Criterion        -Inf
## 
## [[2]]
## Fitting of the distribution ' gamma ' by matching moments 
## Parameters : 
##         estimate
## shape 0.74003981
## rate  0.04794076
## Loglikelihood:  Inf   AIC:  -Inf   BIC:  -Inf
Chi-Squared Test for Gamma Distribution with Shape 0.74003981 and Rate 0.04794076
Pass Error Critical Value
Not rejected 12.19202 15.50731
5.3.2.1.3 Testing for Exponential Distribution

## [[1]]
## Goodness-of-fit statistics
##                              1-mme-exp
## Kolmogorov-Smirnov statistic 0.0764491
## Cramer-von Mises statistic   0.1701794
## Anderson-Darling statistic         Inf
## 
## Goodness-of-fit criteria
##                                1-mme-exp
## Akaike's Information Criterion  1474.275
## Bayesian Information Criterion  1477.558
## 
## [[2]]
## Fitting of the distribution ' exp ' by matching moments 
## Parameters : 
##        estimate
## rate 0.06478132
## Loglikelihood:  -736.1374   AIC:  1474.275   BIC:  1477.558
Chi-Squared Test for Exponential Distribution with Rate 0.06478132
Pass Error Critical Value
Not rejected 10.57594 11.0705

5.3.2.2 ASA 2

## summary statistics
## ------
## min:  1   max:  171 
## median:  14 
## mean:  21.72577 
## estimated sd:  23.44355 
## estimated skewness:  2.526401 
## estimated kurtosis:  12.16981
5.3.2.2.1 Testing for Lognormal Distribution

The lognormal distribution requires numbers above 0. Hence, the shifting in this case is by 14, such that the minimum is 1.

## [[1]]
## Goodness-of-fit statistics
##                              1-mme-lnorm
## Kolmogorov-Smirnov statistic   0.1140782
## Cramer-von Mises statistic     1.1855089
## Anderson-Darling statistic    13.0342018
## 
## Goodness-of-fit criteria
##                                1-mme-lnorm
## Akaike's Information Criterion    4010.304
## Bayesian Information Criterion    4018.673
## 
## [[2]]
## Fitting of the distribution ' lnorm ' by matching moments 
## Parameters : 
##         estimate
## meanlog 2.692986
## sdlog   0.878081
## Loglikelihood:  -2003.152   AIC:  4010.304   BIC:  4018.673
Chi-Squared Test for Lognormal Distribution with MeanLog 2.692986 and SdLog 0.878081
Pass Error Critical Value
Not rejected 11.18576 12.59159
5.3.2.2.2 Testing for Gamma Distribution

## [[1]]
## Goodness-of-fit statistics
##                              1-mme-gamma
## Kolmogorov-Smirnov statistic  0.06179999
## Cramer-von Mises statistic    0.21824158
## Anderson-Darling statistic           Inf
## 
## Goodness-of-fit criteria
##                                1-mme-gamma
## Akaike's Information Criterion        -Inf
## Bayesian Information Criterion        -Inf
## 
## [[2]]
## Fitting of the distribution ' gamma ' by matching moments 
## Parameters : 
##         estimate
## shape 0.78319684
## rate  0.03778855
## Loglikelihood:  Inf   AIC:  -Inf   BIC:  -Inf
Chi-Squared Test for Gamma Distribution with Shape 0.78319684 and Rate 0.03778855
Pass Error Critical Value
Not rejected 4.265961 9.487729
5.3.2.2.3 Testing for Exponential Distribution

## [[1]]
## Goodness-of-fit statistics
##                               1-mme-exp
## Kolmogorov-Smirnov statistic 0.05585831
## Cramer-von Mises statistic   0.27792163
## Anderson-Darling statistic          Inf
## 
## Goodness-of-fit criteria
##                                1-mme-exp
## Akaike's Information Criterion  3912.437
## Bayesian Information Criterion  3916.621
## 
## [[2]]
## Fitting of the distribution ' exp ' by matching moments 
## Parameters : 
##       estimate
## rate 0.0482491
## Loglikelihood:  -1955.218   AIC:  3912.437   BIC:  3916.621
Chi-Squared Test for Exponential Distribution with Rate 0.0482491
Pass Error Critical Value
Not rejected 4.463257 12.59159

5.3.2.3 ASA 3

## summary statistics
## ------
## min:  1   max:  113 
## median:  21 
## mean:  26.55319 
## estimated sd:  21.2979 
## estimated skewness:  1.694304 
## estimated kurtosis:  7.55201
5.3.2.3.1 Testing for Lognormal Distribution

The lognormal distribution requires numbers above 0. Hence, the shifting in this case is by 14, such that the minimum is 1.

## [[1]]
## Goodness-of-fit statistics
##                              1-mme-lnorm
## Kolmogorov-Smirnov statistic   0.1106431
## Cramer-von Mises statistic     0.1018993
## Anderson-Darling statistic     1.5959393
## 
## Goodness-of-fit criteria
##                                1-mme-lnorm
## Akaike's Information Criterion    419.0401
## Bayesian Information Criterion    422.7404
## 
## [[2]]
## Fitting of the distribution ' lnorm ' by matching moments 
## Parameters : 
##          estimate
## meanlog 3.0349670
## sdlog   0.6988318
## Loglikelihood:  -207.52   AIC:  419.0401   BIC:  422.7404
Chi-Squared Test for Lognormal Distribution with MeanLog 3.034967 and SdLog 0.6988318
Pass Error Critical Value
Not rejected 2.577932 7.814728
5.3.2.3.2 Testing for Gamma Distribution

## [[1]]
## Goodness-of-fit statistics
##                              1-mme-gamma
## Kolmogorov-Smirnov statistic  0.05606922
## Cramer-von Mises statistic    0.01871981
## Anderson-Darling statistic           Inf
## 
## Goodness-of-fit criteria
##                                1-mme-gamma
## Akaike's Information Criterion         Inf
## Bayesian Information Criterion         Inf
## 
## [[2]]
## Fitting of the distribution ' gamma ' by matching moments 
## Parameters : 
##         estimate
## shape 1.47081119
## rate  0.05755881
## Loglikelihood:  -Inf   AIC:  Inf   BIC:  Inf
Chi-Squared Test for Gamma Distribution with Shape 1.47081119 and Rate 0.05755881
Pass Error Critical Value
Not rejected 2.209917 9.487729
5.3.2.3.3 Testing for Exponential Distribution

## [[1]]
## Goodness-of-fit statistics
##                              1-mme-exp
## Kolmogorov-Smirnov statistic 0.1239516
## Cramer-von Mises statistic   0.1281206
## Anderson-Darling statistic         Inf
## 
## Goodness-of-fit criteria
##                                1-mme-exp
## Akaike's Information Criterion  400.6316
## Bayesian Information Criterion  402.4818
## 
## [[2]]
## Fitting of the distribution ' exp ' by matching moments 
## Parameters : 
##        estimate
## rate 0.03913405
## Loglikelihood:  -199.3158   AIC:  400.6316   BIC:  402.4818
Chi-Squared Test for Exponential Distribution with Rate 0.03913405
Pass Error Critical Value
Not rejected 6.167518 11.0705

6 Summary and Conclusion

6.1 Planned and Walk-In Appointments

6.1.1 Queue Time

Distributions for each ASA Score
ASA Score Distribution Error Critical Values
1 Lognormal(4.7441447, 0.4887809) 3.790103 9.487729
2 Gamma(4.51396193, 0.02991051) 17.96091 18.30704
3 Gamma(4.4903452, 0.0290352) 13.37659 18.30704

6.1.2 Consult Duration

Distributions for each ASA Score
ASA Score Distribution Error Critical Values
1 Lognormal(2.4525327, 0.8756247) 9.605517 15.50731
2 Exp(0.048233) 5.7713 5.991465
3 Lognormal(3.0657644, 0.8372545) 5.50356 12.59159
Gamma(0.92089709, 0.03126416) 5.842161 5.991465
Exp(0.03394968) 4.483978 7.814728

6.2 Planned Appointments

6.2.1 Queue Time

Distributions for each ASA Score
ASA Score Distribution Error Critical Values
1 Lognormal(4.7362772, 0.4766837) 5.192523 9.487729
2 Gamma(4.6350196, 0.0309402) 20.84311 21.02607
3 Gamma(4.77692467, 0.03145134) 13.61228 14.06714

6.2.2 Consult Duration

Distributions for each ASA Score
ASA Score Distribution Error Critical Values
1 Lognormal(2.456952, 0.874598) 16.66601 16.91898
Gamma(0.7716552, 0.0479119) 13.26114 14.06714
2 Gamma(0.9400198, 0.0453387) 5.40839 5.991465
Exp(0.04823165) 11.10927 12.59159
3 Lognormal(3.0717530, 0.8440134) 4.778335 7.814728
Gamma(0.90118456, 0.03022804) 8.379399 9.487729
Exp(0.03354257) 6.057588 11.0705

6.3 Walk-In Appointments

6.3.1 Queue Time

Distributions for each ASA Score
ASA Score Distribution Error Critical Values
1 Lognormal(4.8384769, 0.5555817) 5.394277 9.487729
Gamma(2.76537723, 0.01876737) 2.535864 7.814728
2 Gamma(3.69937258, 0.02254071) 21.08321 22.36203
Lognormal(4.980962, 0.489148) 18.26444 22.36203
3 Gamma(3.50311077, 0.01894881) 7.495405 14.06714

6.3.2 Consult Duration

Distributions for each ASA Score
ASA Score Distribution Error Critical Values
1 Lognormal(2.4071329, 0.8858606) 0.8173222 7.814728
Gamma(0.74003981, 0.04794076) 12.19202 15.50731
Exp(0.06478132) 10.57594 11.0705
2 Lognormal(2.692986, 0.878081) 11.18576 12.59159
Gamma(0.78319684, 0.03778855) 4.265961 9.487729
Exp(0.0482491) 4.463257 12.59159
3 Lognormal(3.0349670, 0.6988318) 2.577932 7.814728
Gamma(1.47081119, 0.05755881) 2.209917 9.487729
Exp(0.03913405) 6.167518 11.0705